Signal recovery from a few linear measurements of its high-order spectra

The q-th order spectrum is a polynomial of degree q in the entries of a signal x∈C^N, which is invariant under circular shifts of the signal. For q≥3, this polynomial determines the signal uniquely, up to a circular shift, and is called a high-order spectrum. The high-order spectra, and in particular the bispectrum (q=3) and the trispectrum (q=4), play a prominent role in various statistical signal processing and imaging applications, such as phase retrieval and single-particle reconstruction. However, the dimension of the q-th order spectrum is N^q−1, far exceeding the dimension of x, leading to increased computational load and storage requirements. In this work, we show that it is unnecessary to store and process the full high-order spectra: a signal can be uniquely characterized up to symmetries, from only N+1 linear measurements of its high-order spectra. The proof relies on tools from algebraic geometry and is corroborated by numerical experiments.